Hamiltonian Feynman Formula

• Hamiltonian Feynman Formula is a rigorous method approximating quantum propagators via operator product limits and finite-dimensional integrals.
• It unifies continuous and discrete quantum systems through integral kernels, deformation quantization, and sum-over-walks representations.
• The approach underpins numerical simulations and spectral analyses by providing concrete frameworks for Monte Carlo and operator splitting techniques.

The Hamiltonian Feynman formula is a collective term for a family of rigorous mathematical representations that express the propagator or partition function generated by a quantum Hamiltonian as a limit of finite-dimensional algebraic sums, matrix products, or multiple integrals, often motivated by the formalism of path integrals in quantum mechanics. These representations are central to the mathematical theory of quantum dynamics, functional integration, spectral analysis, and quantum statistical mechanics. Originating in Feynman’s path-integral formulation, the Hamiltonian Feynman formula generalizes the sum-over-histories paradigm to both continuous and discrete systems, incorporates developments from deformation quantization, white-noise analysis, Feller process theory, and semigroup product formulas, and provides a foundation for computation, analytic approximation, and Monte Carlo simulation in quantum and stochastic contexts.

1. Formalism and Mathematical Structure

The Hamiltonian Feynman formula refers to representations of the unitary operator $e^{-i t H}$ (or the heat kernel $e^{-\beta H}$) for a self-adjoint (possibly unbounded) Hamiltonian $H$ as either:

• A strong (and sometimes norm) limit of $n$-fold compositions of explicit bounded operators ("quasi-Feynman formulas"/Chernoff products);
• A limit of finite-dimensional integrals with explicit kernels (integral-operator product formulas);
• Combinatorial sums over graph walks when $H$ is a finite matrix (sum-over-walks);
• Phase-space traces and star-exponential formulas in deformation quantization.

A central theme is the approximation,

$e^{-i t H} = \lim_{n\to\infty} [S(t/n)]^n$

where $S(\cdot)$ is a Chernoff-tangent family of bounded, self-adjoint operators closely related to $H$ (Remizov, 2014, Remizov, 2015). For integral-operator cases, each $S(t)$ may itself be an integral kernel, yielding multi-fold integrals as $n\to\infty$ (Butko et al., 2012). These formulas rigorously underpin the otherwise formal path-integral representations, accommodating both real and imaginary time evolution.

2. Deformation Quantization and Star-Exponential Formulation

Within the phase-space (deformation quantization) program, the Hamiltonian Feynman formula is recast entirely in terms of the Moyal star-product and associated star-exponentials. For phase space $e^{-\beta H}$0 with coordinates $e^{-\beta H}$1 and classical Hamiltonian $e^{-\beta H}$2, the star-exponential is defined by

$e^{-\beta H}$3

where $e^{-\beta H}$4 is the $e^{-\beta H}$5-fold Moyal star-product. This object is a phase-space analog of $e^{-\beta H}$6. The partition function is then given as the phase-space trace,

$e^{-\beta H}$7

which, upon Wick rotation ($e^{-\beta H}$8), gives access to the real-time propagator and, in the large $e^{-\beta H}$9 limit, to the ground-state energy $H$0 via

$H$1

As an explicit example, for the 1D harmonic oscillator $H$2, the star-exponential and partition function can be calculated exactly, reproducing the standard results (Berra-Montiel et al., 5 Feb 2025).

This formalism is highly advantageous in that spectral information (eigenvalues, occupation numbers) is encoded in explicit phase-space integrals without requiring operator calculus or coherent state insertions, which is essential in contexts where conventional Hilbert space methods are intractable (Berra-Montiel et al., 5 Feb 2025).

3. Discrete Systems and Sum-over-Walks Formulation

For finite-dimensional systems, the Hamiltonian Feynman formula admits a natural formulation as a sum over walks on a graph. For a Hamiltonian $H$3 in an $H$4-dimensional Hilbert space with orthonormal basis $H$5, the Hamiltonian defines a weighted directed graph $H$6 where $H$7 is the weight of the edge from $H$8 to $H$9. The quantum propagator,

$n$0

expands as a sum over walks $n$1 of length $n$2 from $n$3 to $n$4,

$n$5

where the product is over the sequence of edges in the walk $n$6. The partition function is a sum over all closed walks (Kalev et al., 2024). In the continuum limit (e.g., by lattice discretization and $n$7), such sums approximate the classic path-integral expressions, bridging discrete and continuous quantum systems.

This sum-over-walks formalism provides a transparent algebraic path-integral representation for quantum spin chains, network Hamiltonians, and tight-binding systems. Walk-sums can be sampled numerically and suggest new approximation schemes for network-based quantum dynamics and quantum field theories on lattices (Kalev et al., 2024).

4. Integral Operator and Product Formulas

For generators of Feller semigroups (including variable-coefficient Laplacians and pseudo-differential operators), the Hamiltonian Feynman formula appears as a limit of products of integral operators,

$n$8

with

$n$9

where $H$0 is the pseudodifferential symbol of the Hamiltonian $H$1. This direct approximation provides a bridge to Feynman-Kac formulae and supports both analytic and stochastic simulation approaches to quantum processes and stochastic differential equations (Butko et al., 2012).

The Chernoff theorem guarantees the convergence of such operator products under suitable uniform growth and continuity conditions on the symbol $H$2. With Monte Carlo simulation, these finite-dimensional approximations enable numerical evaluation of evolution semigroups and associated functionals (Butko et al., 2012).

5. Quasi-Feynman Formulas and Operator Splitting

A generalization termed "quasi-Feynman formulas" employs families $H$3 of bounded operators, which are Chernoff-tangent to $H$4, such that

$H$5

with $H$6 and the first-order expansion $H$7. Such constructions hold for a broad class of Hamiltonians on Hilbert spaces, including those composed as finite sums of symmetric or self-adjoint operators (Remizov, 2014, Remizov, 2015). If $H$8 is expressed via integral kernels or multipliers, this limit becomes a multi-fold real or complex integral matching the intuition of time-sliced path integrals.

In the case $H$9, $e^{-i t H} = \lim_{n\to\infty} [S(t/n)]^n$0 can be constructed as a Gaussian convolution composed with a potential multiplier,

$e^{-i t H} = \lim_{n\to\infty} [S(t/n)]^n$1

leading to a quasi-Feynman formula whose limit recovers the standard Feynman-Kac (imaginary-time) path integral (Remizov, 2014).

By comparison, for Hamiltonians equal to a finite sum $e^{-i t H} = \lim_{n\to\infty} [S(t/n)]^n$2, with families $e^{-i t H} = \lim_{n\to\infty} [S(t/n)]^n$3 Chernoff-tangent to $e^{-i t H} = \lim_{n\to\infty} [S(t/n)]^n$4, the main theorem provides

$e^{-i t H} = \lim_{n\to\infty} [S(t/n)]^n$5

with explicit error bounds, unifying Trotter, Chernoff, and Butko–Schilling–Smolyanov product formulas (Remizov, 2015).

6. Connections to Feynman-Kac Theory and Generalizations

The Hamiltonian Feynman formula is fundamentally linked to the Feynman-Kac formula, which gives a probabilistic, path-integral representation for the solution of parabolic and Schrödinger-type equations,

$e^{-i t H} = \lim_{n\to\infty} [S(t/n)]^n$6

where $e^{-i t H} = \lim_{n\to\infty} [S(t/n)]^n$7 is a (possibly jump) process generated by the "free" part of $e^{-i t H} = \lim_{n\to\infty} [S(t/n)]^n$8. The Hamiltonian Feynman formula generalizes this to cases involving nontrivial operator structure, noise, field interaction, and allows description of models such as the spin-boson Hamiltonian and the renormalized relativistic Nelson model via probabilistic integrals over process measures and Fock-space factors (Fröhlich et al., 12 May 2026, Hinrichs et al., 2023).

In the context of deformation quantization, the Hamiltonian Feynman formula connects the spectral decomposition and time-propagation with the phase-space star-exponential, unifying thermal and real-time evolution (Berra-Montiel et al., 5 Feb 2025).

7. Computational and Conceptual Importance

The Hamiltonian Feynman formula admits direct numerical implementation through time-slicing, product of matrices, Monte Carlo sampling of graph walks, and explicit evaluation of Gaussian integrals. It establishes rigorous foundations for the constructive path-integral approach, provides powerful tools for spectral extraction (notably, the ground-state energy via partition function asymptotics), and delivers a manifestly phase-space formulation compatible with quantum-classical transition analyses. By avoiding the need for operator calculus or ill-defined path measures, it supports proof of spectral properties, existence and uniqueness of ground states, and unifies discrete, continuous, and quantum field-theoretic formulations within a common framework (Berra-Montiel et al., 5 Feb 2025, Kalev et al., 2024, Butko et al., 2012, Fröhlich et al., 12 May 2026, Hinrichs et al., 2023, Remizov, 2014, Remizov, 2015).